Differential Geometry of Warped Product Manifolds and Submanifolds

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson–Walker models, are warped product manifolds.

The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson–Walker's and Schwarzschild's.

The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.

The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

Contents:

• Riemannian and Pseudo-Riemannian Manifolds
• Submanifolds
• Warped Product Manifolds
• Robertson-Walker Spacetimes and Schwarzschild Solution
• Contact Metric Manifolds and Submersions
• Kähler and Pseudo-Kähler Manifolds
• Slant Submanifolds
• Generic Submanifolds of Kähler Manifolds
• CR-submanifolds of Kähler Manifolds
• Warped Products in Riemannian and Kähler Manifolds
• Warped Product Submanifolds of Kähler Manifolds
• CR-warped Products in Complex Space Forms
• More on CR-warped Products in Complex Space Forms
• δ-invariants, Submersions and Warped Products
• Warped Products in Nearly Kähler Manifolds
• Warped Products in Para-Kähler Manifolds
• Warped Products in Sasakian Manifolds
• Warped Products in Affine Spaces

Readership: Graduate students and researchers interested in warped product manifolds and submanifolds in geometry, mathematical physics and general relativity.
Key Features:

• Provides an updated and detailed account on results of warped product manifolds and warped product submanifolds